This image and others are part of the Art Exhibit at the 2013 Joint Mathematics Meetings.
You can make your own images like this using Math4Tanabata.
Dubuc and Malik (Convex hull of powers of complex number, trinomial equations and the Farey sequences, Num. Algorithms, 2 #1, pp. 1-32) note that the convex hull of the powers 1, z, z2, z3, . . . of a complex number z, in the interior of the unit disk, is a polygon. They define the color of the number to be the number of vertices of this polygon.
Given epsilon > 0, define the color of the complex number z to be the smallest positive integer n such that |(z - 1).im / (z - 1).re - (zn - 1).im / (zn - 1).re| < epsilon. That is, the slope of the line through z and 1 differs from the slope of the line through zn and 1 by less than epsilon. For this image, epsilon = 0.02 and the highest power of z considered is 24.
Click here for related colorings.
Harriet J. Fell
College of Computer Science
Northeastern University, Boston, MA 02115
Phone: (617) 373-2198
The URL for this document is: http://www.ccs.neu.edu/home/fell/mathGraphics.html